On tilting complexes over blocks covering cyclic blocks

TL;DR

This paper investigates tilting complexes over blocks of group algebras with cyclic defect groups, demonstrating that such blocks are tilting-discrete and that their tilting complexes correspond to those of their subgroups.

Contribution

It proves that blocks covering cyclic defect groups are tilting-discrete and establishes an isomorphism between their tilting complexes and those of the original blocks.

Findings

Blocks with cyclic defect groups are tilting-discrete.

The set of tilting complexes over such blocks is isomorphic to that over the subgroup.

Provides structural insights into tilting theory over group algebra blocks.

Abstract

Let $p$ be a prime number, $k$ an algebraically closed field of characteristic $p$, $\tilde{G}$ a finite group, and $G$ a normal subgroup of $\tilde{G}$ having a $p$-power index in $\tilde{G}$. Moreover let $B$ be a block of $kG$ with a cyclic defect group and $\tilde{B}$ be the unique block of $k\tilde{G}$ covering $B$. We study tilting complexes over the block $\tilde{B}$ and show that the block $\tilde{B}$ is a tilting-discrete algebra. Moreover we show that the set of all tilting complexes over $\tilde{B}$ is isomorphic to that over $B$ as partially ordered sets.